4. (a) (i) Let f(t) be a real valued function defined on (0, 00). Define the...

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4. (a) (i) Let f(t) be a real valued function defined on (0, 00). Define the Laplace transform F(s) of f stating the values of s for which F(s) is defined. (ii) Use your definition in (i) above to determine the Laplace transform of { 2, 0 < t< 5 f(t) : 0, 5 <t < 10 e4t t > 10. (15 marks) (b) Given that y is a function of x, solve the initial value problem: dy (i) 2/y +I cos x, y(T) = 0, %3D dx 15/? (ii) * y tan x = 2x cos r, y 32 (25 marks) 4. (a) (i) Let f(t) be a real valued function defined on (0, 00). Define the Laplace transform F(s) of f stating the values of s for which F(s) is defined. (ii) Use your definition in (i) above to determine the Laplace transform of { 2, 0 < t< 5 f(t) : 0, 5 <t < 10 e4t t > 10. (15 marks) (b) Given that y is a function of x, solve the initial value problem: dy (i) 2/y +I cos x, y(T) = 0, %3D dx 15/? (ii) * y tan x = 2x cos r, y 32 (25 marks)

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